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Farrukh Mukhamedov

Publications and source records attributed to Farrukh Mukhamedov.

At least 91 records · Page 5Linked to original sources
arXiv2014

Renormalization method in $p$-adic $λ$-model on the Cayley tree

In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic $ł$-model on the Cayley tree of order two. We consider generalized $p$-adic quasi Gibbs measures depending on parameter $\r\in\bq_p$, for the $ł$-model. Such measures are constructed by means of certain recurrence equations. These equations define a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed points, which predicts the existence of a phase transition. In the second regime the system has two attractive and one neutral fixed points, which predicts the existence of a quasi phase transition. A main point of this paper is to verify (i.e. rigorously prove) and confirm that the indicated predictions (via dynamical systems point of view) are indeed true. To establish the main result, we employ the methods of $p$-adic analysis, and therefore, our results are not valid in the real setting.

math-ph↗
arXiv2014

Measurable bundles of Banach algebras

In the present paper we investigate Banach--Kantorovich algebras over faithful solid subalgebras of algebras measurable functions. We prove that any Banach--Kantorovich algebra over faithful solid subalgebras of algebra measurable functions represented as a measurable bundle of Banach algebras with vector-valued lifting. We apply such representation to the spectrum of elements Banach--Kantorovich algebras.

math.FA↗
arXiv2014

On non-Archimedean recurrence equations and their applications

In the present paper we study stability of recurrence equations (which in particular case contain a dynamics of rational functions) generated by contractive functions defined on an arbitrary non-Archimedean algebra. Moreover, multirecurrence equations are considered. We also investigate reverse recurrence equations which have application in the study of $p$-adic Gibbs measures. Note that our results also provide the existence of unique solutions of nonlinear functional equations as well.

math.DS↗
arXiv2014

On Volterra and orthoganality preserving quadratic stochastic operators

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. In the present paper, we first give a simple characterization of Volterra QSO in terms of absolutely continuity of discrete measures. Moreover, we provide its generalization in continuous setting. Further, we introduce a notion of orthogonal preserving QSO, and describe such kind of operators defined on two dimensional simplex. It turns out that orthogonal preserving QSOs are permutations of Volterra QSO. The associativity of genetic algebras generated by orthogonal preserving QSO is studied too.

math.DS↗
arXiv2013

On Quantum Markov Chains on Cayley tree III: Ising model

In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found critical temperature coincides with usual critical temperature.

math-ph↗
arXiv2013

Ergodic properties of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

It is known that the Dobrushin's ergodicity coefficient is one of the effective tools to study a behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base (OBSB), and study its properties. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of homogeneous Markov chains. This extends earlier results obtained in case of von Neumann algebras. Such a result allowed to establish a category theorem for uniformly ergodic Markov operators. We find necessary and sufficient conditions for the weak ergodicity of nonhomogeneous discrete Markov chains (NDMC). It is also studied $L$-weak ergodicity of NDMC defined on OBSB. We establish that the chain satisfies $L$-weak ergodicity if and only if it satisfies a modified Doeblin's condition ($\frak{D}_1$-condition). Moreover, several nontrivial examples of NDMC which satisfy the $\frak{D}_1$-condition are provided.

math.FA↗
arXiv2013

On $ξ^{(s)}$-Quadratic Stochastic Operators on two Dimensional simplex and their behavior

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, it was investigated several classes of QSO. In this paper, we study $ξ^{(s)}$--QSO defined on 2D simplex. We first classify $ξ^{(s)}$--QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.

math.DS↗
arXiv2013

On pure quasi quantum quadratic operators of M_2(C)

In the present paper we study quasi quantum quadratic operators (q.q.o) acting on the algebra of $2\times 2$ matrices $M_2(C)$. It is known that a channel is called pure if it sends pure states to pure ones. In this papers, we introduce a weaker condition, called $q$-purity, than purity of the channel. To study $q$-pure channels, we concentrate ourselves to quasi q.q.o. acting on $M_2(C)$. We describe all trace-preserving quasi q.q.o. on $M_2(C)$, which allowed us to prove that if a trace-preserving symmetric quasi q.q.o. such that the corresponding quadratic operator is linear, then its $q$-purity implies its positivity. If a symmetric quasi q.q.o. has a Haar state $τ$, then its corresponding quadratic operator is nonlinear, and it is proved that such $q$-pure symmetric quasi q.q.o. cannot be positive. We think that such a result will allow to check whether a given mapping from $M_2(C)$ to $M_2(C)øM_2(C)$ is pure or not. On the other hand, our study is related to construction of pure quantum nonlinear channels. Moreover, it is also considered that nonlinear dynamics associated with quasi pure q.q.o. may have differen kind of dynamics, i.e. it may behave chaotically or trivially, respectively.

math.DS↗
arXiv2013

On Kadison-Schwarz type quantum quadratic operators on $\bm_2(\mathbb{C})$

In the present paper we study description of Kadison-Schwarz type quantum quadratic operators acting from $\bm_2(\mathbb{C})$ into $\bm_2(\mathbb{C})ø\bm_2(\mathbb{C})$. Note that such kind of operator is a generalization of quantum convolution. By means of such a description we provide an example of q.q.o. which is not a Kadision-Schwartz operator. Moreover, we study dynamics of an associated nonlinear (i.e. quadratic) operators acting on the state space of $\bm_2(\mathbb{C})$.

math.FA↗
arXiv2012

Phase transitions for $P$-adic Potts model on the Cayley tree of order three

In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter $ρ\in\bq_p$. Such a measure is called {\it generalized $p$-adic quasi Gibbs measure}. When $ρ$ equals to $p$-adic exponent, then it coincides with the $p$-adic Gibbs measure. When $ρ=p$, then it coincides with $p$-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of $|ρ|_p$. Namely, in the first regime, one takes $ρ=\exp_p(J)$ for some $J\in\bq_p$, in the second one $|ρ|_p<1$. In each regime, we first find conditions for the existence of generalized $p$-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when $|\r|_p,|q|_p\leq p^{-2}$ we prove the existence of a quasi phase transition. It turns out that if $|\r|_p<|q-1|_p^2<1$ and $\sqrt{-3}\in\bq_p$, then one finds the existence of the strong phase transition.

math-ph↗
arXiv2012

Existence of $P$-adic quasi Gibbs measure for countable state Potts model on the Cayley tree

In the present paper we provide a new construction of measure, called $p$-adic quasi Gibbs measure, for countable state of $p$-adic Potts model on the Cayley tree. Such a construction depends on a parameter $\frak{p}$ and wights. In particular case, i.e. if $\frak{p}=\exp_p$, the defined measure coincides with $p$-adic Gibbs measure. In this paper, under some condition on weights we establish the existence of $p$-adic quasi Gibbs measures associated with the model. Note that this condition does not depend on values of the prime $p$. An analogues fact is not valid when the number of spins is finite.

math-ph↗
arXiv2012

On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces

In this paper, we unify all know iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically $I$-nonexpansive mappings. Note that such a scheme contains as a particular case of the method introduced in [C.E. Chidume, E.U. Ofoedu, \textit{Inter. J. Math. & Math. Sci.} \textbf{2009}(2009) Article ID 615107, 17p]. We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically $I-$nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.

math.FA↗
arXiv2012

On $L_1$-Weak Ergodicity of nonhomogeneous discrete Markov processes and its applications

In the present paper we investigate the $L_1$-weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the $L_1$-weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition for such processes to satisfy the $L_1$-weak ergodicity. Moreover, we apply the obtained results to establish $L_1$-weak ergodicity of discrete time quadratic stochastic processes. As an application of the main result, certain concrete examples are also provided.

math.PR↗
arXiv2012

On cubic equations over $P-$adic field

We provide a solvability criteria for a depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$ are provided. Since $\bbf_p\subset\bq_p,$ we generalize J.-P. Serre's \cite{JPSJ} and Z.H.Sun's \cite{ZHS1,ZHS3} results concerning with depressed cubic equations over the finite field $\bbf_p$. Finally, all depressed cubic equations, for which the Cardano method could be applied, are described and the $p-$adic Cardano formula is provided for those cubic equations.

math.NT↗
arXiv2012

Measurable bundles of $C^*$-dynamical systems and its applications

In the present paper we investigate $L_0$-valued states and Markov operators on $ C^*$-algebras over $L_0$. In particular, we give representations for $L_0$-valued state and Markov operators on $ C^*$ algebras over $L_0$, respectively, as measurable bundles of states and Markov operators. Moreover, we apply the obtained representations to study certain ergodic properties of $ C^*$-dynamical systems over $L_0$.

math.OA↗